If 6A = 11B = 7C: find `A : B : C`.

Options

A. `66:42:77`
B. `77: 42: 66`
C. `42: 77 : 6`
D. `7 : 11 : 6`

Answer

To solve the problem where \(6A = 11B = 7C\) and find the ratio \(A : B : C\), we can follow these steps:

### Step 1: Set a common variable
Let \(6A = 11B = 7C = K\). This means we can express \(A\), \(B\), and \(C\) in terms of \(K\).

### Step 2: Express \(A\), \(B\), and \(C\) in terms of \(K\)
From the equation \(6A = K\), we can express \(A\) as:
\[ A = \frac{K}{6} \]
From the equation \(11B = K\), we can express \(B\) as:
\[ B = \frac{K}{11} \]
From the equation \(7C = K\), we can express \(C\) as:
\[ C = \frac{K}{7} \]

### Step 3: Write the ratio \(A : B : C\)
Now we have:
\[ A : B : C = \frac{K}{6} : \frac{K}{11} : \frac{K}{7} \]

### Step 4: Eliminate \(K\) from the ratio
Since \(K\) is common in all three terms, we can simplify the ratio by removing \(K\):
\[ A : B : C = \frac{1}{6} : \frac{1}{11} : \frac{1}{7} \]

### Step 5: Find the LCM of the denominators
To combine these fractions into a single ratio, we need to find the least common multiple (LCM) of the denominators 6, 11, and 7. Since these numbers are distinct and have no common factors, the LCM is:
\[ \text{LCM} = 6 \times 11 \times 7 = 462 \]

### Step 6: Convert each term to have the same denominator
Now, we convert each term in the ratio to have the same denominator of 462:
- For \(A\):
\[ A = \frac{1}{6} = \frac{77}{462} \]
- For \(B\):
\[ B = \frac{1}{11} = \frac{42}{462} \]
- For \(C\):
\[ C = \frac{1}{7} = \frac{66}{462} \]

### Step 7: Write the final ratio
Now we can write the final ratio:
\[ A : B : C = 77 : 42 : 66 \]

### Final Answer
Thus, the ratio \(A : B : C\) is:
\[ \boxed{77 : 42 : 66} \]